It is intuitively clear that the generalized Bayes framework could completely resolve the model misspecification if the learning rate was chosen as \(\eta ^\star = (\sigma /\sigma ^\star )^2\) . More formally, [1]} show that, if the learning rate is no larger than this ratio, the generalized Bayes posterior will enjoy fast root-\(n\)  rate convergence properties, while [2]} argue that learning rate equal to the ratio in order to achieve exact coverage of credible sets. In any case, of course, one cannot make this learning rate choice in practice because it depends on the unknown value of true variance. But this intuition tells us what the different learning rate selection methods' target should be.
